HYACINTHOS 26546

 

[Antreas P. Hatzipolakis]:

 

The natural question now is:

Let A'B'C', A"B"C" be the cevian, circumcevian triangle of a point P, resp.

Let A2, B2, C2 be the isogonal conjugates of A', B', C' wrt A"B"C", resp.

Which is the locus of P such that A"B"C", A2B2C2 are perspective?

The entire plane?


[César Lozada]:

 

Locus for perspectivity: the entire plane.

 

For P=u:v:w (trilinears), the perspector Z2(P) is:

 

Z2(P) = a*(u^3*(-a^2+b^2+c^2)*v*w*(a*( b*w+c*v)+u*b*c)+(b^4+c^4-a^4)* v^2*u^2*w^2-a*u*v^2*w^2*((a^2- b^2)*b*v-(c^2-a^2)*c*w)+b*c*(( v^2+w^2)*u^4*b*c+a*u^3*(b*w^3+ c*v^3)+u^2*v*w*(b^2*w^2+c^2*v^ 2)-a^2*v^3*w^3)) : :

 

or, for P=x : y : z (barycentrics)

Z2(P) = x^3*(-a^2+b^2+c^2)*y*z*(a^2*( z*b^2+y*c^2)+x*b^2*c^2)+x^2*(( b^2*z^2+c^2*y^2)*b^2*c^2*x^2+( b^4*z^3+c^4*y^3)*a^2*x+(b^4*z^ 2+c^4*y^2)*a^2*y*z)+(b^4+c^4- a^4)*a^2*x^2*y^2*z^2-a^4*x*y^ 2*z^2*((a^2-b^2)*y-(c^2-a^2)* z)-a^6*y^3*z^3 : :

 

Z2( P on the circumcircle) = P

 

If P is in the infinity, Z2(P) lies on the circumcircle and it is the reflection of the isogonal-of-P on the line {O, orthopoint-of-P}.

 

Some ETC-pairs for P in the infinity: (30,476), (511,805), (512,805), (513,901), (514,927), (515,1309), (516,927), (517,901), (518,6078), (519,6079), (520,6080), (521,6081), (522,1309), (523,476), (524,6082), (525,2867),…

 

 

Other ETC pairs (P,Z2(P)): (1,3), (2,6031), (3,20), (4,4), (5,14097), (6,353)

 

Examples:

 

Z2( X(7) ) = 4*a^6-8*(b+c)*a^5+11*b*c*a^4+( b+c)*(8*b^2-11*b*c+8*c^2)*a^3- (4*b^2+9*b*c+4*c^2)*(b-c)^2*a^ 2-(b^2-c^2)*(b-c)*b*c*a+2*b*c* (b-c)^4 : :  (barycentrics)

= (4*s^4+8*(5*R^2-SW)*s^2+5*R*S* s-2*(16*R^2-r^2-2*SW)*SW)*X(7) +4*(S+2*s*R)*(S-s*R)*X(1155)

= on line {7, 1155}

= [ 2.726145614955648, 2.38844268976661, 0.728906182089505 ]

 

Z2( X(8) ) = (p^3*(16*p^3-16*p^2*q-5*p+20* q)-(9*q^2+7)*p^2+2*(q^2+2)*q* p-q^2)/p^2 : : (trilinears)

where p=sin(A/2), q=cos((B-C)/2)

= on line {8, 1319}

= [ 12.250956385605930, 4.92689920087082, -5.424553296667387 ]

 

 

César Lozada